Which of the following equations has no real solutions? A. 8(x − 5) = 8x − 40 B. 8x − 5 = 40x − 5 C. 8x − 5 = 8x − 5 D. 8(x + 5) = 8(x + 9)

The equation that has no real solutions is B. 8x − 5 = 40x − 5.

Here's why:

Let's solve each equation:

A. 8(x − 5) = 8x − 40 This simplifies to 8x - 40 = 8x - 40, which is always true for any real number x.

B. 8x − 5 = 40x − 5 Subtract 8x from both sides to get -5 = 32x - 5. Then add 5 to both sides to get 0 = 32x. This implies that x = 0, but if you substitute x = 0 back into the original equation, you get -5 = -5, which is a contradiction. Therefore, this equation has no real solutions.

C. 8x − 5 = 8x − 5 This is always true for any real number x.

D. 8(x + 5) = 8(x + 9) This simplifies to 8x + 40 = 8x + 72. Subtract 8x from both sides to get 40 = 72, which is not true. Therefore, this equation has no real solutions. However, the original equation is not equivalent to the simplified equation, so there was a mistake in the simplification. If you distribute the 8 on the left side of the original equation, you get 8x + 40 = 8x + 72, which is always true for any real number x. Therefore, this equation does have real solutions.